A lithographic apparatus is a machine that applies a desired pattern onto a substrate, usually onto a target portion of the substrate. A lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). In that instance, a patterning device, which is alternatively referred to as a mask or a reticle, may be used to generate a circuit pattern to be formed on an individual layer of the IC. This pattern can be transferred onto a target portion (e.g., including part of, one, or several dies) on a substrate (e.g., a silicon wafer). Transfer of the pattern is typically via imaging onto a layer of radiation-sensitive material (resist) provided on the substrate. In general, a single substrate will contain a network of adjacent target portions that are successively patterned. Known lithographic apparatus include so-called steppers, in which each target portion is irradiated by exposing an entire pattern onto the target portion at one time, and so-called scanners, in which each target portion is irradiated by scanning the pattern through a radiation beam in a given direction (the “scanning”-direction) while synchronously scanning the substrate parallel or anti-parallel to this direction. It is also possible to transfer the pattern from the patterning device to the substrate by imprinting the pattern onto the substrate.
In order to monitor the lithographic process, it is desirable to measure parameters of the patterned substrate, for example, the overlay error between successive layers formed in or on the patterned substrate. There are various techniques for making measurements of the microscopic structures formed in lithographic processes, including the use of scanning electron microscopes and various specialized tools. One form of specialized inspection tool is a scatterometer in which a beam of radiation is directed onto a target on the surface of the substrate and properties of the scattered or reflected beam are measured. By comparing the properties of the beam before and after it has been reflected or scattered by the substrate, the properties of the substrate may be determined. This may be done, for example, by comparing the reflected beam with data stored in a library of known measurements associated with known substrate properties. Two main types of scatterometer are known. Spectroscopic scatterometers direct a broadband radiation beam onto the substrate and measure the spectrum (e.g., intensity as a function of wavelength) of the radiation scattered into a particular narrow angular range. Angularly resolved scatterometers use a monochromatic radiation beam and measure the intensity of the scattered radiation as a function of angle.
Models are often used to simulate results from scatterometers or spectrometers. To determine a critical dimension, a modeled signal may be matched to a measured signal. Within the model there are many parameters (e.g., the thickness or reflectivity of layers of the substrate) which may be varied to generate a modeled spectrum which matches the measured signal. With many different parameters varying freely the matching process is extremely time consuming to run. Too many free parameters may result in an unstable matching process or erroneous set of parameters due to the fact that there may exist more than one combination of these parameters that have virtually equal modeled spectra. Consequently, many of the parameters are often fixed while just a few are varied. However, it may be difficult to determine which parameters may be left free while the others are fixed.
There may be some correlation between the impact of different parameters on the modeled spectrum and present methods of determining which parameters to leave free involve the use of a cross-correlation matrix. A value for each of the parameters is selected and a base spectrum generated. A parameter is varied by a small amount, another spectrum is generated and the change of the spectrum is determined. This is repeated for each of the parameters and the resulting spectra changes between the different parameters are compared to generate the cross-correlation matrix. If a high correlation between two parameters is found at most one of them should be left free. However, while this correlation matrix supports the selection of free and fixed parameters, the use is limited since it may not show the impact of correlation between the spectral change for more than two parameters, it does not give any indication of the impact of the noise in the measured signal on the model with a specific free parameter selection nor does it show the impact of errors in the value of the fixed parameters on the free parameters during the matching process or on the quality of the match.
Furthermore the cross-correlation matrix provides no information about the effect of converting a free parameter to a fixed parameter. Changing a free parameter to a fixed parameter or vice versa could have an unexpected effect on other parameters.